Nonzero-sum Games Research Articles

A Formal Model on How the Clean Development Mechanism of the Kyoto Protocol Produces Perverse Effects

One of the main environmental problems in the contemporary society is global warming. In 1997, as a result of negotiations on that issue, several countries signed the Kyoto Protocol that institutes the Clean Development Mechanism (CDM) as an ancillary instrument to decrease the global emissions of greenhouse gases (GHG). We argue that the current CDM design, contrary to many suppositions, produces the perverse effect of increasing the global emissions of GHG. In this study we propose a formal model to support our central argument. Initially, we use game theory to formalize our explanation about how the two players (Kyoto Protocol ANNEX1 and NON-ANNEX1 countries) cooperate by such mechanism, in a non-zero sum game. Eventually, microeconomic theory is used to model the dynamic of interactions of Carbon Tradable Offsets (CTO) buyers and sellers, thereby demonstrating that the cooperation between players produces the perverse effect. This explanatory study used rational choice theory and gathered data in books, scientific articles and official documents purposing to figure out how the CDM works and to support the argument. The main conclusion of our model is that after the CTO demanded quantity be supplied, NON-ANNEX1 countries will use the revenue from CTO markets to increase their productive capacity, so that there will be a turn over in the engagement for reducing GHG emissions by these countries, and the perverse effect of global increasing at such emissions level will follow.

Nash equilibrium point for one kind of stochastic nonzero-sum game problem and BSDEs

In this Note, we deal with one kind of stochastic nonzero-sum differential game problem for N players. Using the theory of backward stochastic differential equations and Malliavin calculus, we give the explicit form of a Nash equilibrium point. To cite this article: J.-P. Lepeltier et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

PARAMETER ANALYSIS OF A ZERO-SUM DIFFERENTIAL GAME APPROXIMATING A NONZERO-SUM GAME

In this paper we consider a zero-sum differential game in feedback setting with one state coordinate and finite time horizon. The problem is characterized by two parameters. This game arises as an approximation to a nonzero-sum game. Using analytical and numerical methods we solve the HJBI equation and give the description of the optimal feedback controls for both players. These feedbacks are given in terms of switching curves and singular universal lines. We also determine the change (bifurcations) of the optimal phase portrait of the game depending upon the parameters. In conclusion, we show that for some values of the parameters the solution of the considered zero-sum game leads to the exact solution of the corresponding nonzero-sum game, and for the other values of the parameters it supplies just some approximation.

Multi-Robot Searching using Game-Theory Based Approach

This paper proposes a game-theory based approach in a multi–target searching using a multi-robot system in a dynamic environment. It is assumed that a rough priori probability map of the targets' distribution within the environment is given. To consider the interaction between the robots, a dynamic-programming equation is proposed to estimate the utility function for each robot. Based on this utility function, a cooperative nonzero-sum game is generated, where both pure Nash Equilibrium and mixed-strategy Equilibrium solutions are presented to achieve an optimal overall robot behaviors. A special consideration has been taken to improve the real-time performance of the game-theory based approach. Several mechanisms, such as event-driven discretization, one-step dynamic programming, and decision buffer, have been proposed to reduce the computational complexity. The main advantage of the algorithm lies in its real-time capabilities whilst being efficient and robust to dynamic environments.

Equilibrium and Pareto-optimality in noisy discrete duels with an arbitrary number of actions

We study a nonzero-sum game of two players that is a generalization of the antagonistic noisy duel of discrete type. The game is considered from the point of view of various criteria of optimality. We prove the existence of e-equilibrium situations and show that the e-equilibrium strategies that we found are e-maxmin. Conditions under which the equilibrium plays are Pareto-optimal are given.

The complexity of equilibria: Hardness results for economies via a correspondence with games

We give a reduction from any two-player game to a special case of the Leontief exchange economy, with the property that the Nash equilibria of the game and the equilibria of the market are in one-to-one correspondence. Our reduction exposes a computational hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies is at least as hard as finding a Nash equilibrium for two-player nonzero sum games, a problem recently proven to be P P A D -complete. As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [I. Gilboa, E. Zemel, Nash and correlated equilibria: Some complexity considerations, Games and Economic Behavior 1 (1989) 80–93]. In particular, among other results, we show that it is N P -hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium. Perhaps more importantly, we also prove that it is N P -hard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known P P A D -completeness result of [C.H. Papadimitriou, On the complexity of the parity argument and other inefficient proofs of existence, Journal of Computer and System Sciences 48 (1994) 498–532], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwer’s Fixed Point Theorem.

Stochastic Learning Solution for Distributed Discrete Power Control Game in Wireless Data Networks

Distributed power control is an important issue in wireless networks. Recently, noncooperative game theory has been applied to investigate interesting solutions to this problem. The majority of these studies assumes that the transmitter power level can take values in a continuous domain. However, recent trends such as the GSM standard and Qualcomm's proposal to the IS-95 standard use a finite number of discretized power levels. This motivates the need to investigate solutions for distributed discrete power control which is the primary objective of this paper. We first note that, by simply discretizing, the previously proposed continuous power adaptation techniques will not suffice. This is because a simple discretization does not guarantee convergence and uniqueness. We propose two probabilistic power adaptation algorithms and analyze their theoretical properties along with the numerical behavior. The distributed discrete power control problem is formulated as an N-person, nonzero sum game. In this game, each user evaluates a power strategy by computing a utility value. This evaluation is performed using a stochastic iterative procedures. We approximate the discrete power control iterations by an equivalent ordinary differential equation to prove that the proposed stochastic learning power control algorithm converges to a stable Nash equilibrium. Conditions when more than one stable Nash equilibrium or even only mixed equilibrium may exist are also studied. Experimental results are presented for several cases and compared with the continuous power level adaptation solutions.

Teaching Nonzero Sum Games Using a Diagrammatic Determination of Equilibria

We present a diagrammatic method that allows the determination of all Nash equilibria of 2 × M nonzero sum games, extending thus the well known diagrammatic techniques for 2 × M zero sum and 2 × 2 nonzero sum games. We show its appropriateness for teaching purposes by analyzing modified versions of the prisoners' dilemma, the battle of the sexes, as well as of the zero sum game of matching pennies. We then use the method to give simple proofs for the existence of Nash equilibria in all 2 × M games, for both the nonzero sum (Nash existence theorem) and the zero sum case (Minimax theorem). We also prove in the same spirit the remarkable general fact that in a nondegenerate 2 × M nonzero sum game there is an odd number of equilibria.

NASH EQUILIBRIA IN UNCONSTRAINED STOCHASTIC GAMES OF RESOURCE EXTRACTION

A class of nonzero-sum symmetric stochastic games of capital accumulation/resource extraction is considered. It is shown that Nash equilibria in the games with some natural constraints are also equilibrium solutions in unconstrained games and dominate in the Pareto sense an equilibrium leading to exhausting the entire resource stock in the first period of the game.

Learning game strategy design through iterated Prisoner's Dilemma

The article investigates games strategies on the basis of experiments with iterated Prisoner's Dilemma, a classical non-zero sum game. The objective is to determine which strategies have the best chance of winning. Although some strategies, like tit-for-tat, emerge as better than others in some cases, it appears that there is no overall winning strategy, but that success or failure of individual strategies depends upon the strategies adopted by a population of opponents. Therefore, the winning strategy will change dynamically, and will need to be determined while the game in progress. Based on the results of this work, a strategy engine for games development is proposed, and a future development of strategy middleware is discussed.

Asymptotic ε-Nash Equilibrium for 2nd Order Two-Player Nonzero-sum Singular LQ Games with Decentralized Control

We propose a new equilibrium concept: asymptotic ε-Nash equilibrium for 2***nd order two-player nonzero-sum games where each player has a control-free cost functional quadratic in the system states over an infinite horizon and each player's control strategy is constrained to be continuous linear state feedback. Based on each player's singular control problem, the asymptotic ε-Nash equilibrium implemented by partial state feedback is constructed and the feedback gains can be found by solving a group of algebraic equations which involves the system coefficients and weighting matrices in the cost functionals. As an illustration of the theories discussed in this paper, a numerical example is given where the partial state feedback gains can be found explicitly in terms of the system coefficients and weighting matrices in the cost functionals.

Selection of a correlated equilibrium in Markov stopping games

This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players’ decisions according to some optimality criterion. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the best choice problem are given. Several concepts of criteria for selecting a correlated equilibrium are used.

Nondominated equilibrium solutions of a multiobjective two-person nonzero-sum game in extensive form and corresponding mathematical programming problem

In most of studies on multiobjective noncooperative games, games are represented in normal form and a solution concept of Pareto equilibrium solutions which is an extension of Nash equilibrium solu...

Concurrent games with tail objectives

We study infinite stochastic games played by two players over a finite state space, with objectives specified by sets of infinite traces. The games are concurrent (players make moves simultaneously and independently), stochastic (the next state is determined by a probability distribution that depends on the current state and chosen moves of the players) and infinite (proceed for an infinite number of rounds). The analysis of concurrent stochastic games can be classified into: quantitative analysis, analyzing the optimum value of the game and ε -optimal strategies that ensure values within ε of the optimum value; and qualitative analysis, analyzing the set of states with optimum value 1 and ε -optimal strategies for the states with optimum value 1. We consider concurrent games with tail objectives, i.e., objectives that are independent of the finite-prefix of traces, and show that the class of tail objectives is strictly richer than that of the ω -regular objectives. We develop new proof techniques to extend several properties of concurrent games with ω -regular objectives to concurrent games with tail objectives. We prove the positive limit-one property for tail objectives. The positive limit-one property states that for all concurrent games if the optimum value for a player is positive for a tail objective Φ at some state, then there is a state where the optimum value is 1 for the player for the objective Φ . We also show that the optimum values of zero-sum (strictly conflicting objectives) games with tail objectives can be related to equilibrium values of nonzero-sum (not strictly conflicting objectives) games with simpler reachability objectives. A consequence of our analysis presents a polynomial time reduction of the quantitative analysis of tail objectives to the qualitative analysis for the subclass of one-player stochastic games (Markov decision processes).

Nondominated Equilibrium Solutions of a Multiobjective Two-Person Nonzero-Sum Game and Corresponding Mathematical Programming Problem

With reference to a multiobjective two-person nonzero-sum game, we define nondominated equilibrium solutions and provide a necessary and sufficient condition for a pair of mixed strategies to be a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. We give a numerical example and demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.

KNOWLEDGE DIASPORAS FOR DEVELOPMENT

This paper deals with the contribution of diaspora knowledge networks to the development of home countries as a major potential brain gain. Although some questions were recently raised about the consistency, viability and efficiency of the diaspora option, the latest studies tend to validate its role in the development of knowledge intensive activities at home. The Asian experience (China and India), the current expansion of their highly skilled expatriate associations, their real intensity of activity as well as their responsiveness to policy factors, confirm the diaspora option. The case of the development of the Information Technology industry in India demonstrates the active involvement of the diaspora. Its integration into the global market has gone far beyond the sole effect of reputation enhancer, a minimal positive role often attributed to the diaspora. In fact, the active role of the Indian diaspora, a creative mediator rather than a passive intermediary in these processes, has changed strategic balance and geopolicy. The diaspora factor is indeed part of the knowledge transfer to the Indian IT industry that helps it to move up on the value chain. The circulation exemplified by diasporas justifies shifting from binary/dichotomical explanations (host versus home countries benefits), to the analysis of mediation processes at work in non-zero sum game analysis.

Extra-proximal methods for solving two-person nonzero-sum games

We consider two-person nonzero-sum game, both in the classical form and in the form of a game with coupled variables. An extra-proximal approach for finding the game’s solutions is suggested and justified. We provide our algorithm with an analysis of its convergence.

A game version of the Cowan–Zabczyk–Bruss’ problem

The paper deals with the continuous-time two person non-zero sum game extension of the no information secretary problem. The objects appear according to the compound Poisson process and each player can choose only one applicant. If both players would like to select the same one, then the priority is assigned randomly. The aim of the players is to choose the best candidate. A construction of Nash equilibria for such game is presented. The extension of the game with randomized stopping times is taken into account. The Nash values for such extension are obtained. Analysis of the solutions for different priority defining lotteries is given.

Experimental games: a review.

A fresh view of the strategies employed by individuals, groups, or social institutions in conflict situations has been stimulated by the mathematics of game theory. This has provided us with norms concerning reasonable decision policies to follow under various circumstances, but not with information on actual behavior. Something about this can be learned from experiments in which subjects resolve dilemmas involving opposition of interests. Such research is rapidly becoming voluminous, but an overview of it can still be crammed into one article. Such a review must deal with 2- and with n-person games, with zero-sum and nonzero-sum games, with games where payoffs are known and unknown, and with simulations which involve games.

Effectiveness of the Nash strategies in competitive multi-team target assignment problems

The Nash strategy in game theory has often been criticized as being ineffective in competitive multi-team target assignment problems, especially when compared with other simplistic strategies such as the random or greedy targeting strategies. This criticism arises from the fact that the Nash strategies may yield unpredictable results when paired with non-Nash strategies in non-zero sum games. In addition, the Nash equilibrium is generally more difficult to compute than strategies which do not attempt to anticipate the strategy of the other side. The authors seek to show that in multi-team target assignment problems the Nash strategy is superior to such simplistic strategies while also remaining computationally feasible. To demonstrate this point, an attrition model was considered, consisting of two teams of nonhomogeneous fighting units simultaneously targeting each other and compare the outcomes when various combinations of four targeting strategies are used on each side. The four strategies are: 1) the random strategy where each unit selects its target randomly, 2) the unit greedy strategy where each unit chooses the target that optimizes its own performance only, 3) the team optimal strategy where the units coordinate their choice of targets so as to optimize the overall team performance while ignoring the possible strategy choices by the other team, and 4) the team Nash strategy, calculated under the assumption that the other team is also using a Nash strategy. Because the computational requirements for calculating the team Nash strategies may become unfeasibly large, an efficient method was discussed for approximating the Nash strategies using a neighborhood search algorithm called unit level team resource allocation (ULTRA). The results were compared for all 16 possible combinations of these four targeting strategies and show that for each team the Nash strategy outperforms all other strategies irrespective of the strategy employed by the other team